# Instrument-Armed Bandits

## TL;DR

This paper introduces the instrument-armed bandit (IAB) model, extending classic multi-armed bandits to settings with noncompliance, relevant for human trials, analyzing regret notions, and developing new algorithms with theoretical guarantees.

## Contribution

The paper formalizes the IAB problem, explores different regret notions, and proposes new algorithms with proven regret bounds tailored for noncompliance scenarios.

## Key findings

- Standard MAB algorithms cannot achieve sublinear regret in IAB settings.
- New algorithms for IAB achieve better regret bounds.
- Numerical experiments compare IAB algorithms to classical MAB methods.

## Abstract

We extend the classic multi-armed bandit (MAB) model to the setting of noncompliance, where the arm pull is a mere instrument and the treatment applied may differ from it, which gives rise to the instrument-armed bandit (IAB) problem. The IAB setting is relevant whenever the experimental units are human since free will, ethics, and the law may prohibit unrestricted or forced application of treatment. In particular, the setting is relevant in bandit models of dynamic clinical trials and other controlled trials on human interventions. Nonetheless, the setting has not been fully investigate in the bandit literature. We show that there are various and divergent notions of regret in this setting, all of which coincide only in the classic MAB setting. We characterize the behavior of these regrets and analyze standard MAB algorithms. We argue for a particular kind of regret that captures the causal effect of treatments but show that standard MAB algorithms cannot achieve sublinear control on this regret. Instead, we develop new algorithms for the IAB problem, prove new regret bounds for them, and compare them to standard MAB algorithms in numerical examples.